Ballistic fingerprinting has been used as evidence in US courts and elsewhere. In this method found bullets are compared based on markings the firing gun left on them. This should allow to prove which gun a certain bullet was fired from.

I've heard about several forensic techniques that were widely used but were not substantiated by evidence, e.g. a certain way of determining if a fire is arson. In the linked article it is also mentioned that some lawmakers want to abolish the database of ballistic fingerprints as it is claimed to be ineffective.

How good is the evidence for ballistic fingerprinting? Specifically,

How often are the projectiles undamaged enough to allow ballistic fingerprinting?

How high is the false positive rate, projectiles identified as belonging to a certain gun that weren't fired from it?

How high is the false negative rate, how often is no match found if the correct projectile and gun are compared?

Unfortunately, they do not answer any of the specific questions you raised in your post, and the studies they reference do not all appear to be available online. However, I think a few things are made clear.

First, the specific type of bullet and gun is important to the results obtained. Soft lead bullets are much harder to recover intact and identify than copper-jacketed bullets. Caseless bullets or bullets where the case is not recovered are harder to identify, since the case is often more robustly identifiable. This is made clear in section 3 of the Technical Evaluation.

Second, section 3-E of the National Academies report explicitly found that "The validity of the fundamental assumptions of uniqueness and reproducibility of firearms-related toolmarks has not yet been fully demonstrated." (bold theirs) and that "Conclusions drawn in firearms identification should not be made to imply the presence of a firm statistical basis when none has been demonstrated." On the other hand, they also report that at a minimal level, "there is at least some 'signal' that may be detected."

On a personal note, I found the National Academies report to be quite convincing, and recommend those interested in more details to read it.